Proof Roof

Monday, 9 February 2015

A few months ago on his blog Terry Tao explained how one could, by analogy with the theory of graph limits, replace the explicit use of arithmetic regularity with a soft device which he calls an additive limit. Roughly speaking, the additive limit, or Kronecker factor, of a sequence of finite abelian groups $G_i$ is a compact quotient of the ultraproduct $\prod G_i$ which controls the convolutions. The result is that many theorems from additive combinatorics, such as Roth's theorem, which are usually proved using quantitative tools like Fourier analysis can instead be proved using soft tools more like the Lebesgue differentiation theorem.

The purpose of this post is to extend Tao's construction to the nonabelian setting. Tao already stated in his post that this should be possible, so one could say that this is just an exercise in nonabelian Fourier analysis. On the other hand the proof in the nonabelian setting more or less forces a more categorical point of view, so certain points of this exercise are instructive.

1. Measurable Bohr compactification

$\def\B{\text{Bohr}}\def\Ba{\text{Baire}}$Given any topological group $G$ there is a compact group $\B(G)$, called the Bohr compactification of $G$, such every continuous homomorphism from $G$ to a compact group $K$ factors uniquely through $\B(G)$. One can think of $\B(G)$ as the 'largest' compact group in which $G$ has dense image. We need a variant of this definition for groups $G$ endowed only with a $\sigma$-algebra instead of a topology.

By a measurable group we mean a group $G$ together with a $\sigma$-algebra $\Sigma$ of subsets of $G$. Note that we do not make any measurability assumptions about the group operation or even the left- or right-shifts, though certainly it would be sensible to do so in other contexts. For us the only role of $\Sigma$ is to distinguish among all homomorphisms the measurable homomorphisms. The analogue of Bohr compactification for measurable groups is given by the following theorem.

Theorem 1 (Existence of measurable Bohr compactification): For every measurable group $G$ there is a compact group $\B_m(G)$ together with a $(\Sigma,\Ba)$-measurable homomorphism $\pi:G\to \B_m(G)$ such that every $(\Sigma,\Ba)$-measurable homomorphism from $G$ to a compact group $K$ factors uniquely as the composition of $\pi:G\to \B_m(G)$ and a continuous homomorphism $\B_m(G)\to K$.

In category-theoretic terms $\B_m$ is a left adjoint to the functor $\Ba$ from compact groups to measurable groups which replaces a group's topology with its Baire $\sigma$-algebra. By the adjoint functor theorem it suffices to check that the functor $\Ba$ is continuous, which boils down to the following lemma. Incidentally the analogue of this lemma fails for the Borel $\sigma$-algebra, which is why we must consider the Baire $\sigma$-algebra instead.

Lemma 2: For compact Hausdorff spaces $X_i$ we have $\Ba(\prod X_i) = \prod \Ba(X_i)$. In words, the Baire $\sigma$-algebra of the product is the product of the Baire $\sigma$-algebras.

Proof: The containment $\prod \Ba(X_i) \subset \Ba(\prod X_i)$ is immediate from the defintions. To prove the opposite containment it suffices to check that every continuous function $f:\prod X_i \to \mathbf{R}$ is measurable with respect to $\prod\Ba(X_i)$. This is certainly true of functions $f$ which depend on only finitely many coordinates, and thus for all continuous functions $f$ by the Stone-Weierstrass theorem.$\square$

Those who, like me, are not used to thinking in terms of the adjoint functor theorem will appreciate a more pedestrian proof of Theorem 1. To this end, let $\mathcal{K}$ be the set of all pairs $(f,K)$, where $K$ is a compact group and $f:G\to K$ is a $(\Sigma,\Ba)$-measurable homomorphism with $f(G)$ dense in $K$. Then if$$\pi:G\to\prod_{(f,K)\in\mathcal{K}} K$$ is the diagonal map then $\pi:G\to\overline{\pi(G)}$ is the Bohr compactification of $G$. Indeed, the measurability of $\pi$ follows from Lemma 2, and the universal property is essentially obvious: given a $(\Sigma,\Ba)$-measurable homomorphism $f:G\to K$ with $K$ compact, the pair $(f,\overline{f(G)})$ appears in $\mathcal{K}$, so we have continuous maps$$\overline{\pi(G)}\to\prod_{(g,L)\in\mathcal{K}} L \to \overline{f(G)}\to K$$whose composition $h$ satisfies $h\pi = f$; moreover $h$ is unique because $\pi(G)$ is dense in $\overline{\pi(G)}$.

Alternatively, by the Peter-Weyl theorem, $\B_m(G)$ can be defined as the inverse limit of all measurable finite-dimensional unitary representations of $G$.

It is natural to ask what relation the measurable Bohr compactification $\B_m$ bears to the usual Bohr compactification $\B$. In particular, is $\B$ just the composition $\B_m\circ\Ba$? Clearly $\B(G) = \B_m(\Ba(G))$ if and only if every measurable homomorphism $f:G\to K$ from $G$ to a compact group $K$ is continuous. This follows from Steinhaus's theorem if $G$ is locally compact, but it certainly fails in general, for instance for $G=\mathbf{Q}$ with the topology inherited from $\mathbf{R}$.

2. The Bohr compactification of an ultrafinite group

Let $G_1,G_2,\dots$ be a sequence of finite groups and let $p\in \beta\mathbf{N} \setminus\mathbf{N}$ be a nonprincipal ultrafilter. We form the ultraproduct $G=\prod_{n\to p} G_n$ and make it into a measurable group by giving it the Loeb $\sigma$-algebra $\mathcal{L}_G$, the $\sigma$-algebra generated by internal sets $\prod_{n\to p} A_n$, where $A_n\subset G_n$. We define the Loeb measure $\mu_G$ on internal sets $\prod_{n\to p}A_n$ by putting$$\mu_G\left(\prod_{n\to p}A_n\right)=\text{st}\lim_{n\to p} |A_n|/|G_n|,$$and we define $\mu_G$ on $\mathcal{L}_G$ by extension.

While the group operation $G\times G\to G$ is not generally measurable with respect to the product $\sigma$-algebra $\mathcal{L}_G\times\mathcal{L}_G$, it is measurable with respect to the larger $\sigma$-algebra $\mathcal{L}_{G\times G}$. Moreover this latter $\sigma$-algebra is still 'product-like' in the sense that all $\mathcal{L}_{G\times G}$-measurable $f:G\times G\to\mathbf{R}_{\geq0}$ obey Fubini's theorem, so we have a sensibly defined convolution operation $L^1(G)\times L^1(G)\to L^1(G)$ given $$f*g(x) = \int f(y)g(y^{-1}x)\,d\mu_G(y).$$

Now consider the Bohr compactification $\B_m(G)$ of $G$. The first thing to notice is that $\pi_*\mu_G$ is a $\pi(G)$-invariant Baire probability measure on $\B_m(G)$. Since $\pi(G)$ is dense in $\B_m(G)$ we conclude that $\pi_*\mu_G$ is in fact $\B_m(G)$-invariant, so by the uniqueness of Haar measure we must have$$\pi_*\mu_G=\mu_{\B_m(G)}.$$

In the remainder of this section we relate the two convolution algebras $L^2(G)$ and $L^2(\B_m(G))$. Given $f\in L^2(\B_m(G))$ we can form the pullback $\pi^*f = f\circ\pi$. Since$$ \|f\circ\pi\|_{L^2(G)}^2 = \int |f\circ\pi|^2\,d\mu_G = \int |f|^2 \,d\mu_{\B_m(G)} = \|f\|_{L^2(\B_m(G))}^2,$$we see that $\pi^*f$ is a well defined element of $L^2(G)$, and in fact $\pi^*$ defines an isometric embedding$$\pi^*:L^2(\B_m(G))\to L^2(G).$$In the other direction we have the pushforward$$\pi_*:L^2(G)\to L^2(\B_m(G)),$$ defined as the adjoint of $\pi_*$. The identities $$\pi_*\pi^*f=f\quad\text{for }f\in L^2(\B_m(G)),$$$$\pi^*(f*g)=\pi^*f*\pi^*f\quad\text{for }f,g\in L^2(\B_m(G)),$$$$\pi_*(f*g)=\pi_*f*\pi_*g\quad\text{for }f,g\in L^2(G)$$are readily verified. For instance, the last of these is verified by the following computation, valid for $h\in L^2(\B_m(G))$ by $\mathcal{L}_{G\times G}$-Fubini:$$\langle h, \pi_*(f*g)\rangle = \langle \pi^* h, f*g\rangle = \int h(\pi(xy)) f(x)g(y)\,d\mu_G(x)\,d\mu_G(y)$$$$=\int h(x'y') \pi_*f(x')\pi_*g(y')\,d\mu_{\B_m(G)}(x')\,d\mu_{\B_m(G)}(y') = \langle h,\pi_*f*\pi_*g\rangle.$$We can summarise the situation as an isometric Banach algebra isomorphism$$L^2(G) \cong L^2(\B_m(G))\oplus \ker\pi_*.$$

The following theorem asserts that in fact $\B_m(G)$ alone determines convolutions in $G$, and thus $\B_m(G)$ will more generally control all 'first-order configurations' in $G$.

Theorem 3: We have $(\ker\pi_*)*L^2(G)=0$. Thus all convolutions in $L^2(G)$ can be computed in $L^2(\B_m(G))$, in the sense that$$f*g=\pi^*(\pi_*f*\pi_*g)$$for all $f,g\in L^2(G)$.

From an additive combinatorics point of view, nonabelian groups obey a structure versus randomness principle: the asymptotic behaviour with respect to linear configurations can usually be described as some combination of abelian and random-like behaviour. Following Gowers, we call a sequence of finite groups $(G_n)$ quasirandom if the least dimension of a nontrivial representation of $G_n$ tends to infinity with $n$. For example for $n\geq 7$ every nontrivial representation of the alternating group $\text{Alt}(n)$ has dimension at least $n-1$, so the sequence $(\text{Alt}(n))$ is quasirandom.

Theorem 5: The ultrafinite group $G_p=\prod_{n\to p} G_n$ has trivial Bohr compactification if and only if the least dimension of a nontrivial representation of $G_n$ tends to infinity as $n\to p$. In particular, $(G_n)$ is quasirandom if and only if $\B_m(G_p)$ is trivial for every $p\in\beta\mathbf{N}\setminus\mathbf{N}$.

First we need a simple lemma.

Lemma 6: Let $G$ and $H$ be groups with $G$ finite and $f:G\to H$ a map which satisfies $f(xy)=f(x)f(y)$ for $1-o(1)$ of the pairs $(x,y)\in G^2$. Then there is a homomorphism $h:G\to H$ such that $f(x)=h(x)$ for $1-o(1)$ of the points $x\in G$.

Proof: For every $x\in G$ and for $1-o(1)$ of the pairs $(y,z)\in G^2$ we have$$f(xyz)f(yz)^{-1} = f(xy)f(z)(f(y)f(z))^{-1} = f(xy)f(y)^{-1},$$so for each $x\in G$ there is a unique $h(x)\in H$ such that$$h(x) = f(xy)f(y)^{-1}$$ for $1-o(1)$ of the points $y\in G$. Clearly $h(x)=f(x)$ for $1-o(1)$ of the points $x\in G$, and for $x_1,x_2\in G$ we have$$h(x_1)h(x_2)^{-1} = f(x_1y)f(y)^{-1}f(y)f(x_2 y)^{-1} = f(x_1y)f(x_2y)^{-1}=h(x_1x_2^{-1})$$for $1-o(1)$ of the points $y\in G$, in particular for at least one $y\in G$, so $h$ is a homomorphism.$\square$

Proof of Theorem 5: Suppose we have a sequence of nontrivial homomorphisms $f_n:G_n\to U(d)$ for all $n$ on some neighbourhood of $p$. Then $(f_n)$ induces a measurable homomorphism $f:G_p\to U(d)$, and since $U(d)$ has no small subgroups the induced homomorphism $f:G_p\to U(d)$ will also be nontrivial, so $\B_m(G_p)$ must be nontrivial.

Conversely suppose the least dimension of a nontrivial representation of $(G_n)$ tends to infinity as $n$ tends to $p$, and let $f:G_p\to U(d)$ be a measurable homomorphism. By a countable saturation argument there is an internal function $g:G_p\to U(d)$, say $g=\st\lim_{n\to p}g_n$, such that $f=g$ almost everywhere. Then $g_n$ satisfies $g_n(xy)=g_n(x)g_n(y)$ for $1-o(1)$ of the pairs $(x,y)\in G_n$, so by the lemma there is a homomorphism $h_n:G_n\to U(d)$ such that $g_n(x) = h_n(x)$ for $1-o(1)$ of the points $x\in G_n$. But by assumption any such homomorphism $h_n$ must be trivial for $n$ near enough to $p$, so we must have $g=1$ almost everywhere, so $f=1$ almost everywhere. Moreover since $f(x) = f(xy)f(y)^{-1}$ we must in fact have $f=1$ identically. Since the Peter-Weyl theorem implies that $\B_m(G_p)$ is an inverse limit of matrix groups this implies that $\B_m(G_p)$ is trivial.$\square$

Equations are generally easy to solve in quasirandom groups. We illustrate this point with the following theorem.

Theorem 7: Let $(G_n)$ be quasirandom and let $\epsilon>0$. Then there exists $n_0$ such that if $n\geq n_0$ and $A_n\subset G_n$ has density $|A_n|/|G_n|\geq\epsilon>0$ then we can find $x,y,z\in A_n$ with $xy=z$.

Proof: Let $p\in\beta\mathbf{N}\setminus\mathbf{N}$, let $G=\prod_{n\to p}G_n$, and let $f$ be the internal function $\st\lim_{n\to p} 1_{A_n}$. Then $\int f\,d\mu_G \geq \epsilon$, so if $\pi:G\to\B_m(G)$ is the Bohr compactification then $\int\pi_* f\,d\mu_{\B_m(G)}\geq\epsilon$. But by the previous theorem $\B_m(G)$ is trivial, so $\pi_* f$ is a constant $\geq\epsilon$, so by Theorem 3 we have$$\langle f*f,f\rangle_{L^2(G)} = \langle \pi^*(\pi_*f*\pi_*f),f\rangle_{L^2(G)} = \langle \pi_*f*\pi_*f,\pi_*f\rangle_{L^2(\B_m(G))} \geq \epsilon^3.$$In other words the number of pairs $(x,y)\in G_n^2$ such that $x,y,xy\in A_n$ is at least $(\epsilon^3-o(1))|G_n|$ as $n\to p$, but since $p$ was arbitrary this must hold as $n\to\infty$.$\square$

Here is another nice criterion for quasirandomness, which can be found in Gowers's original paper: $(G_n)$ is not quasirandom if and only if the groups $G_n$ have nontrivial abelian quotients or nontrivial small quotients. In our setup we can write this the following way.

Theorem 8: Let $G$ be an ultrafinite group. Then one of the following three alternatives hold:
1. $\B_m(G)$ is trivial.
2. $\B_m(G)$ has a nontrivial abelian quotient.
3. $\B_m(G)$ has a nontrivial finite quotient.

Proof: Suppose $G=\prod_{n\to p} G_n$. By the previous theorem if $\B_m(G)$ is nontrivial then the groups $G_n$ have bounded-dimensional nontrivial representations $\pi_n:G_n\to U(d)$ as $n\to p$. By Jordan's theorem, $\pi_n(G_n)$ has a normal abelian subgroup $A_n$ of bounded index. If $A_n=\pi_n(G_n)$ as $n\to p$ then 2 holds, while if $A_n<\pi_n(G_n)$ as $n\to p$ then 3 holds.$\square$

Using this theorem one can prove a sort of converse to Theorem 7. If $(G_n)$ is not quasirandom then there are arbitrarily large $n$ and product-free subsets $A_n\subset G_n$ of density bounded away from $0$. We leave the details to the reader.

4. Roth's theorem

As an application of group limits we can prove the following version of Roth's theorem.

Theorem 9: Let $G$ be a finite group on which the squaring map $s:y\mapsto y^2$ is $O(1)$-to-$1$. Let $A\subset G$ be a subset of density $|A|/|G|\geq\epsilon>0$. Then there are $\gtrsim_\epsilon |G|^2$ solutions to $y^2=xz$ in $A$. Equivalently, there are $\gtrsim_\epsilon|G|^2$ pairs $(a,b)\in G^2$ such that $a,ab,bab\in A$.

We chose to count configurations of the form $(a,ab,bab)$ precisely because they are alternatively described by the rather simple equation $y^2=xz$. If instead we chose to count the more "obvious" nonabelian analgues of three-term arithmetic progressions, namely configurations of the form $(a,ab,ab^2)$, then we would be counting solutions to the more complicated equation $z = yx^{-1}y$. The problem is that the count of these configurations is not obviously controlled by convolutions, so we can't easily transport the problem to the Bohr compactification. In fact the situation is delicate and not completely understood: see for example this paper of Tao for the case of $\text{SL}_d(F)$.

Friday, 6 February 2015

Theorem (Hofmann and Russo): If $G$ is a compact group of positive commuting probability then the FC-center $F(G)$ is an open subgroup of $G$ with finite-index center $Z(F(G))$.

(I actually stated this theorem incorrectly previously, asserting the conclusion $G=F(G)$ as well; this is clearly false in general, for instance for $G=O(2)$.)

Here the FC-center of a group is the subgroup of elements with finitely many conjugates. In general a group is called FC if each of its elements has finitely many conjugates, and BFC if its elements have boundedly many conjugates. A theorem of Bernhard Neumann states that a group $G$ is BFC if and only if $[G,G]$ is finite.

I noticed today that one can prove this theorem rather easily by adapting the proof of Peter Neumann's theorem that a finite group with commuting probability bounded away from $0$ is small-by-abelian-by-small. Some parts of the argument below are present in scattered places in the above two papers, but I repeat them for completeness.

Proof: Let $\mu$ be the normalised Haar measure of $G$, and suppose that$$\mu(\{(x,y):xy=yx\})\geq\epsilon.$$Let $X_n$ be the set of elements in $G$ with at most $n$ conjugates. Then $X_n$ is closed, since any element $x$ with at least $n+1$ distinct conjugates $g_i^{-1}xg_i$ has a neighbourhood $U$ such that for all $u\in U$ the points $g_i^{-1}ug_i$ are distinct. Since$$\mu(\{(x,y):xy=yx\}) = \int 1/|x^G| \,d\mu(x) \leq \mu(X_n) + 1/n,$$we see that $\mu(X_n)\geq\epsilon/2$ for all $n\geq 2/\epsilon$. This implies that the group $H_n$ generated by $X_n$ is generated in at most $6/\epsilon$ steps, i.e., $H_n = X_n^{\lfloor 6/\epsilon\rfloor}$, which implies that $H_n$ is an open BFC subgroup of $G$. Since $(H_n)$ is an increasing sequence of finite-index subgroups it must terminate with some subgroup $F$, and in fact $F$ must be the FC-center of $G$. This proves that $F(G)$ is an open BFC subgroup of $G$.

In particular in its own right $F$ is a compact group with $[F,F]$ finite. Since the commutator map $[,]:F\times F\to[F,F]$ is a continuous map to a discrete set satisfying $[F,1]=1$ there must be a neighbourhood $U$ of $1$ such that $[F,U]=1$. This implies that $Z(F)$ is open, hence of finite-index in $F$.$\square$

For me, the Hofmann-Russo theorem is a negative result: it states that commuting probability does not extend in an interesting way to the category of compact groups. To be more specific we have the following corollary.

Corollary: If $G$ is a compact group of commuting probability $p>0$ then there is a finite group $H$ also of commuting probability $p$.

We need a simple lemma before proving the corollary.

Lemma: For each $n>0$ there is a finite group $K_n$ of commuting probability $1/n$.

Proof: If $n$ is odd then $D_n$ has commuting probability $(n+3)/(4n)$. We can use this formula alone and induction on $n$ to define appropriate groups $K_n$. Take $K_1=D_1$ and $K_2=D_3$. If $n>2$ is even take $K_n = K_2\times K_{n/2}$. If $n = 4k+1>2$ take $K_n = K_{k+1}\times D_n$. If $n=4k+3>2$ take $K_n = K_{k+1}\times D_{3n}$.$\square$

An isoclinism between two groups $G$ and $H$ is a pair of isomorphisms $G/Z(G)\to H/Z(H)$ and $[G,G]\to [H,H]$ which together respect the commutator map $[,]:G/Z(G)\times G/Z(G)\to[G,G]$. Clearly isoclinism preserves commuting probability. A basic theorem on isoclinism, due to Hall, is that every group $G$ is isoclinic to a stem group, a group $H$ satisfying $Z(H)\leq [H,H]$.

Proof of corollary: By the theorem the FC-center $F$ of $G$ has finite-index, say $n$, and moreover $F$ has finite-index center $Z(F)$ and therefore finite commutator subgroup $[F,F]$. Let $E$ be a stem group isoclinic to $F$. Then $E$ and $F$ have the same commuting probability, and $E$ is finite since $E/Z(E) \cong F/Z(F)$, $[E,E]\cong [F,F]$, and $Z(E)\leq [E,E]$, so we can take $H=K_{n^2}\times E$.$\square$

Thursday, 29 January 2015

The fundamental objects of study in higher-order Fourier analysis are nilmanifolds, or in other words spaces given as a quotient $G/\Gamma$ of a connected nilpotent Lie group $G$ by a discrete cocompact subgroup $\Gamma$. Starting with Furstenberg's work on Szemeredi's theorem and the multiple recurrence theorem, work by Host and Kra, Green and Tao, and several others has gradually established that nilmanifolds control higher-order linear configurations in the same way that the circle, as in the Hardy-Littlewood circle method, controls first-order linear configurations.

Of basic importantance in the study of nilmanifolds is equidistribution: one needs to know when the sequence $g^n x$ equidistributes and when it is trapped inside a subnilmanifold. It turns out that this problem was already studied by Leon Green in the 60s. To describe the theorem note first that the abelianisation map $G\to G/G_2$ induces a map from $G/\Gamma$ to a torus $G/(G_2\Gamma)$ which respects the action of $G$, and recall that equidistribution on tori is well understood by Weyl's criterion. Leon Green's beautiful theorem then states that $g^n x$ equidistributes in the nilmanifold if and only if its image in the torus $G/(G_2\Gamma)$ equidistributes.

Today at our miniseminar Aled Walker showed us Parry's nice proof of this theorem, which is more elementary than Green's original proof. During the talk there was some discussion about the importantance of various hypotheses such as 'simply connected' and 'Lie'. It turns out that the proof works rather generally for connected locally compact nilpotent groups, so I thought I would record the proof here with minimal hypotheses. The meat of the argument is exactly as in Aled's talk and, presumably, Parry's paper.

Let $G$ be an arbitrary locally compact connected nilpotent group, say with lower central series$$G=G_1\geq G_2 \geq\cdots\geq G_s\geq G_{s+1}=1,$$and let $\Gamma\leq G$ be a closed cocompact subgroup. Under these conditions the Haar measure $\mu_G$ of $G$ induces a $G$-invariant probability measure $\mu_{G/\Gamma}$ on $G/\Gamma$. We say that $x_n\in G/\Gamma$ is equidistributed if for every $f\in C(G/\Gamma)$ we have$$\frac{1}{N}\sum_{n=0}^{N-1} f(x_n) \to \int f \,d\mu_{G/\Gamma}.$$We fix our attention on the sequence$$x_n = g^n x$$for some $g\in G$ and $x\in G/\Gamma$. As before we have an abelianisation map$$\pi: G/\Gamma\to G/G_2\Gamma$$from the $G$-space $G/\Gamma$ to the compact abelian group $G/G_2\Gamma$. We define equidistribution on $G/G_2\Gamma$ similarly. The theorem is then the following.

Theorem: (Leon Green) For $g\in G$ and $x\in G/\Gamma$ the following are equivalent.
1. The sequence $g^n x$ is equidistributed in $G/\Gamma$.
2. The sequence $\pi(g^n x)$ is equidistributed in $G/G_2\Gamma$.
3. The orbit of $\pi(g)$ is dense in $G/G_2\Gamma$.
4. $\chi(\pi(g))\neq 0$ for every nontrivial character $\chi:G/G_2\Gamma\to\mathbf{R}/\mathbf{Z}$.

Item 1 above trivially implies every other item. The implication 4$\implies$3 (a generalised Kronecker theorem) follows by pulling back any nontrivial character of $(G/G_2\Gamma)/\overline{\langle\pi(g)\rangle}$. The implication 3$\implies$2 (a generalised Weyl theorem) follows from the observation that every weak* limit point of the sequence of measures$$ \frac{1}{N}\sum_{n=0}^{N-1} \delta_{\pi(g^n x)}$$must be shift-invariant and thus equal to the Haar measure. So the interesting content of the theorem is 2$\implies$1.

A word about the relation to ergodicity: By the ergodic theorem the left shift $\tau_g:x\mapsto gx$ is ergodic if and only if for almost every $x$ the sequence $g^n x$ equidistributes; on the other hand $\tau_g$ is uniquely ergodic, i.e., the only $\tau_g$-invariant measure is the given one, if and only if for every $x$ the sequence $g^n x$ equidistributes. Thus to prove the theorem above we must not only prove that $\tau_g$ is ergodic but that it is uniquely ergodic. Fortunately one can prove these two properties are equivalent in this case.

Lemma: If $\tau_g:G/\Gamma\to G/\Gamma$ is ergodic then it's uniquely ergodic.

Proof: (Furstenberg) By the ergodic theorem the set $A$ of $\mu_{G/\Gamma}$-generic points, in other words points $x$ for which$$\frac{1}{N}\sum_{n=0}^{N-1} f(g^n x) \to \int f \,d\mu_{G/\Gamma}$$for every $f\in C(G/\Gamma)$, has $\mu_{G/\Gamma}$-measure $1$, and clearly if $x\in A$ and $c\in G_s$ then $xc\in A$, so $A = p^{-1}(p(A))$, where $p$ is the projection of $G/\Gamma$ onto $G/G_s\Gamma$.

Now let $\mu'$ be any $\tau_g$-invariant ergodic measure. By induction we may assume that $\tau_g:G/G_s\Gamma\to G/G_s\Gamma$ is uniquely ergodic, so we must have $p_*\mu' = p_*\mu_{G/\Gamma}$, so$$\mu'(A) = p_*\mu'(p(A)) = p_*\mu_{G/\Gamma}(p(A)) = \mu_{G/\Gamma}(A) = 1.$$But by the ergodic theorem the set of $\mu'$-generic points must also have $\mu'$-measure $1$, so there must be some point which is both $\mu_{G/\Gamma}$- and $\mu'$-generic, and this implies that $\mu'=\mu_{G/\Gamma}$.$\square$

We need one more preliminary lemma about topological groups before we really get started on the proof.

Lemma: If $H$ and $K$ are connected subgroups of some ambient topological group then $[H,K]$ is also connected.

Proof: Since $(h,k)\mapsto [h,k]=h^{-1}k^{-1}hk$ is continuous certainly $C = \{[h,k]:h\in H,k\in K\}$ is connected, so $C^n = CC\cdots C$ is also connected, so because $1\in C^n$ for all $n$ we see that $[H,K]=\bigcup_{n=1}^\infty C^n$ is connected.$\square$

Thus if $G$ is connected then every term $G_1,G_2,G_3,\dots$ in the lower central series of $G$ is connected.

Proof of Leon Green's theorem: As noted it suffices to prove that $\tau_g$ acts ergodically on $G/\Gamma$ whenever it acts ergodically on $G/G_2\Gamma$. By induction we may assume that $\tau_g$ acts ergodically on $G/G_s\Gamma$. So suppose that $f\in L^2(G/\Gamma)$ is $\tau_g$-invariant. By decomposing $L^2(G/\Gamma)$ as a $\overline{G_s\Gamma}/\Gamma$-space we may assume that $f$ obeys$$f(cx)=\gamma(c)f(x)\quad(c\in G_s, x\in G/\Gamma)$$for some character $\gamma:G_s\Gamma/\Gamma\to S^1$. In particular $|f|$ is both $G_s$-invariant and $\tau_g$-invariant, so it factors through a $\tau_g$-invariant function $G/G_s\Gamma\to\mathbf{R}$, so it must be constant, say $1$. Moreover for every $b\in G_{s-1}$ the function$$\Delta_bf(x) = f(bx)\overline{f(x)}$$is $G_s$-invariant, and also a $\tau_g$ eigenvector:$$\Delta_bf(gx) = \gamma([b,g])\Delta_bf(x).$$By integrating this equation we find that either $\gamma([b,g])=1$, so $\Delta_bf$ is constant, or $\int\Delta_bf \,d\mu_{G/\Gamma}= 0$, so either way we have$$\int \Delta_bf\,d\mu_{G/\Gamma}\in \{0\}\cup S^1.$$But since $\int\Delta_bf\,d\mu_{G/\Gamma}$ is a continuous function of $b$ and equal to $1$ when $b=1$ we must have $\gamma([b,g])=1$ for all sufficiently small $b$, and thus for all $b$ by connectedness of $G_{s-1}$ and the identity$$[b_1b_2,g]=[b_1,g][b_2,g].$$Thus setting $\gamma(b)=\Delta_bf$ extends $\gamma$ to a homomorphism $G_{s-1}\to S^1$. In fact we can extend $\gamma$ still further to a function $G\to D_1$, where $D_1$ is the unit disc in $\mathbf{C}$, by setting$$\gamma(a) = \int \Delta_af\,d\mu_{G/\Gamma}.$$Now if $a\in G$ and $b\in G_{s-1}$ then$$\gamma(ba) = \int f(bax) \overline{f(x)}\,d\mu_{G/\Gamma} = \int \gamma(b)f(ax)\overline{f(x)}\,d\mu_{G/\Gamma}=\gamma(b)\gamma(a),$$and$$\gamma(ab) = \int f(abx)\overline{f(x)}\,d\mu_{G/\Gamma} = \int f(ax) \overline{f(b^{-1}x)}\,d\mu_{G/\Gamma} = \int f(ax) \overline{\gamma(b^{-1})}\overline{f(x)}\,d\mu_{G/\Gamma} = \gamma(b)\gamma(a),$$so$$\gamma(b)\gamma(a)=\gamma(ab)=\gamma(ba[a,b]) = \gamma(ba)\gamma([a,b])=\gamma(b)\gamma(a)\gamma([a,b]).$$Since $|\gamma(b)|=1$ we can cancel $\gamma(b)$, so$$\gamma(a)(\gamma([a,b])-1) = 0.$$Finally observe that $\gamma(a)$ is a continuous function of $a$, and $\gamma(1)=1$, so we must have $\gamma([a,b])=1$ for all sufficiently small $a$, and thus by connectedness of $G$ and the identity$$[a_1a_2,b]=[a_1,b][a_2,b]$$we must have $\gamma([a,b])=1$ identically. But this implies that $\gamma$ vanishes on all $s$-term commutators and thus on all of $G_s$, so in fact $f$ factors through $G/G_s\Gamma$, so it must be constant.$\square$

A remark is in order about the possibility that some of the groups $G_i$ and $G_i\Gamma$ are not closed. This should not matter. One could either read the above proof as it is written, noting carefully that I never said groups should be Hausdorff, or, what's similar, instead modify it so that whenever you quotient by a group $H$ you instead quotient by the group $\overline{H}$.

Embarrassingly, it's difficult to come up with a non-Lie group to which this generalised Leon Green's theorem applies. It seems that many natural candidates have the property that $G$ is not connected but $G/\Gamma$ is: for example consider$$\left(\begin{array}{ccc}1&\mathbf{R}\times\mathbf{Q}_2&\mathbf{R}\times\mathbf{Q}_2\\0&1&\mathbf{R}\times\mathbf{Q}_2\\0&0&1\end{array}\right)/\left(\begin{array}{ccc}1&\mathbf{Z}[1/2]&\mathbf{Z}[1/2]\\0&1&\mathbf{Z}[1/2]\\0&0&1\end{array}\right).$$So it would be interesting to know whether the theorem extends to such a case. Or perhaps there are no interesting non-Lie groups for this theorem, which would be a bit of a let down.